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Properties

Label 1-107-107.48-r0-0-0
Degree $1$
Conductor $107$
Sign $-0.545 - 0.837i$
Analytic cond. $0.496905$
Root an. cond. $0.496905$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.320 − 0.947i)2-s + (0.674 − 0.737i)3-s + (−0.794 + 0.606i)4-s + (0.375 − 0.926i)5-s + (−0.915 − 0.403i)6-s + (0.992 + 0.118i)7-s + (0.829 + 0.558i)8-s + (−0.0887 − 0.996i)9-s + (−0.998 − 0.0592i)10-s + (−0.630 + 0.776i)11-s + (−0.0887 + 0.996i)12-s + (0.147 − 0.989i)13-s + (−0.205 − 0.978i)14-s + (−0.430 − 0.902i)15-s + (0.263 − 0.964i)16-s + (0.0296 + 0.999i)17-s + ⋯
L(s)  = 1  + (−0.320 − 0.947i)2-s + (0.674 − 0.737i)3-s + (−0.794 + 0.606i)4-s + (0.375 − 0.926i)5-s + (−0.915 − 0.403i)6-s + (0.992 + 0.118i)7-s + (0.829 + 0.558i)8-s + (−0.0887 − 0.996i)9-s + (−0.998 − 0.0592i)10-s + (−0.630 + 0.776i)11-s + (−0.0887 + 0.996i)12-s + (0.147 − 0.989i)13-s + (−0.205 − 0.978i)14-s + (−0.430 − 0.902i)15-s + (0.263 − 0.964i)16-s + (0.0296 + 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.545 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.545 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(107\)
Sign: $-0.545 - 0.837i$
Analytic conductor: \(0.496905\)
Root analytic conductor: \(0.496905\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{107} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 107,\ (0:\ ),\ -0.545 - 0.837i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5257477516 - 0.9700843929i\)
\(L(\frac12)\) \(\approx\) \(0.5257477516 - 0.9700843929i\)
\(L(1)\) \(\approx\) \(0.8040394657 - 0.7560004586i\)
\(L(1)\) \(\approx\) \(0.8040394657 - 0.7560004586i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad107 \( 1 \)
good2 \( 1 + (-0.320 - 0.947i)T \)
3 \( 1 + (0.674 - 0.737i)T \)
5 \( 1 + (0.375 - 0.926i)T \)
7 \( 1 + (0.992 + 0.118i)T \)
11 \( 1 + (-0.630 + 0.776i)T \)
13 \( 1 + (0.147 - 0.989i)T \)
17 \( 1 + (0.0296 + 0.999i)T \)
19 \( 1 + (-0.956 + 0.292i)T \)
23 \( 1 + (-0.205 + 0.978i)T \)
29 \( 1 + (-0.533 + 0.845i)T \)
31 \( 1 + (0.582 - 0.812i)T \)
37 \( 1 + (-0.984 - 0.176i)T \)
41 \( 1 + (0.889 - 0.456i)T \)
43 \( 1 + (0.375 + 0.926i)T \)
47 \( 1 + (0.889 + 0.456i)T \)
53 \( 1 + (-0.320 + 0.947i)T \)
59 \( 1 + (-0.533 - 0.845i)T \)
61 \( 1 + (0.992 - 0.118i)T \)
67 \( 1 + (0.829 - 0.558i)T \)
71 \( 1 + (0.674 + 0.737i)T \)
73 \( 1 + (0.937 - 0.348i)T \)
79 \( 1 + (-0.861 + 0.508i)T \)
83 \( 1 + (0.972 - 0.234i)T \)
89 \( 1 + (-0.915 + 0.403i)T \)
97 \( 1 + (-0.998 - 0.0592i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−30.22106756305205626844085294057, −28.656268209483219455314033425809, −27.43908882134198998977789010362, −26.633404762275850355469690068121, −26.147087894200418497530050641203, −25.01727110726965690494568970920, −24.04613082831790073788368358085, −22.80752648556988450626645722875, −21.663708590567451983819659970, −20.86139450617605822773312952941, −19.17348927184739990493177248833, −18.47117807726241158064400300465, −17.23981911302371827026730225312, −16.11006360000392664548428211305, −15.09984248178458319095789804138, −14.15716083420301047381055985074, −13.68786787335919558092789101965, −11.1175601475517495887522056714, −10.2884644818284913827293249184, −9.00820109514020164050422545178, −8.062371214412944620215614385519, −6.82983359490346506829501536390, −5.34234988896683282635149314884, −4.13470349325384689445114146723, −2.32288849252184122943655505922, 1.37735981651585476007870269764, 2.30078684446266344072202094276, 4.061720330492741533795995078043, 5.491199178644841171755047342312, 7.7835080842956350796158527851, 8.35617553744417313346182043124, 9.52245840180590202976449572619, 10.831988066192814252130311898041, 12.41354723129172430503307212953, 12.82358994300882385398881352312, 13.98126484550951787150035590079, 15.2648378249132869020310237595, 17.3431871527459588120858846246, 17.713542239505452621605499123396, 18.92519429776716998201275325944, 20.08967167483498541416052634559, 20.71383232146060077413679563473, 21.48488109250140618609414259404, 23.21204002511532681774248864790, 24.15340878447569015861395783662, 25.34122882439391574492383092722, 26.09597664186657213402420844809, 27.65810413867372337244980793019, 28.15748775221285271930264430728, 29.48021746988522000313779159628

Graph of the $Z$-function along the critical line